A334079 Total area of all triangles such that p + q = 2*n, p < q (p, q prime), with base (q - p) and height p.

0, 0, 0, 3, 6, 5, 12, 30, 34, 42, 54, 81, 72, 78, 126, 78, 168, 224, 84, 201, 332, 279, 252, 571, 462, 339, 760, 441, 522, 1002, 312, 915, 1194, 282, 1116, 1410, 810, 1233, 1778, 1092, 1206, 2332, 1218, 1212, 3396, 1536, 1404, 3017, 1572, 2250, 3988, 2055

Offset

1,4

Links

Table of n, a(n) for n=1..52.

Eric Weisstein's World of Mathematics, Goldbach Partition

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..n-1} i * (n-i) * c(i) * c(2*n-i), where c is the prime characteristic (A010051).

Example

a(4) = 3; 2*4 = 8 has one Goldbach partition into distinct parts (5,3). The area of the triangle is then (5 - 3)*3/2 = 3.
a(8) = 30; 2*8 = 16 has two Goldbach partitions into distinct parts, (13,3) and (11,5). The sum of the areas of the two triangles is (13 - 3)*3/2 + (11 - 5)*5/2 = 15 + 15 = 30.

Mathematica

Table[Sum[i (n - i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n - 1}], {n, 60}]

Crossrefs

Cf. A010051.

Sequence in context: A048724 A292682 A334748 * A328403 A199126 A247569

Adjacent sequences: A334076 A334077 A334078 * A334080 A334081 A334082

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 13 2020

Status

approved